How to List All Outcomes in Sample Space for Coins, Dice, and Cards
The concept of sample space in Maths plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding sample spaces is essential for solving probability questions, whether you are preparing for board exams, Olympiads, or competitive tests like JEE.
What Is Sample Space in Maths?
A sample space in maths is defined as the set of all possible outcomes of a random experiment. You’ll find this concept applied in areas such as probability theory, statistics, and even computer science. For example, when flipping a coin, the possible outcomes are {Head, Tail} and the sample space is written as S = {H, T}.
Key Formula for Sample Space in Maths
Here’s the standard formula: \( \text{Number of outcomes in sample space} = n_1 \times n_2 \times ... \times n_k \)
For certain cases like tossing n coins, the formula for sample space is \( 2^n \).
Properties and Types of Sample Space
| Type | Description | Example |
|---|---|---|
| Finite (Discrete) | Sample space with a countable number of outcomes | Tossing a die: S = {1, 2, 3, 4, 5, 6} |
| Infinite | Sample space with infinitely many outcomes | Measuring time until it rains: S = {all positive real numbers} |
| Continuous | Outcome takes values in a continuous interval | Length of a rod between 2 cm and 3 cm |
Sample Space Examples
- Flipping a single coin: S = {H, T}
- Tossing two coins: S = {HH, HT, TH, TT}
- Rolling a die: S = {1, 2, 3, 4, 5, 6}
- Rolling two dice: S = {(1,1), (1,2), ..., (6,6)} (36 outcomes)
- Drawing a card from a deck: S = {all 52 cards}
How to Find the Sample Space
- Identify the experiment (e.g., tossing coins, rolling dice).
- List every possible result. For n coins, use \( 2^n \); for two dice, use 6 × 6 = 36.
- Write the set S using curly brackets {} and separate outcomes by commas.
- For complex cases, use diagrams or tables if it helps to visualize.
Sample Space in Probability Calculations
In probability, the sample space forms the denominator in probability formulas. If the event is a subset of the sample space, then
\( P(\text{event}) = \frac{\text{Number of favourable outcomes}}{\text{Total outcomes in sample space}} \)
For example, the probability of rolling an even number on a die is \( P(\text{even}) = \frac{3}{6} = 0.5 \), since S = {1,2,3,4,5,6} and there are three even numbers (2,4,6).
Step-by-Step Illustration
Problem: What is the sample space when flipping three coins?
1. Start by noting each coin has 2 outcomes: H (Head), T (Tail)2. For 3 coins, total outcomes = \( 2^3 = 8 \)
3. List all possible results:
4. So, the sample space S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}
Speed Trick or Vedic Shortcut
Here’s a quick trick: If you’re tossing n coins, use \( 2^n \) straight away to find the sample space size instead of listing each outcome. It saves time during board and competitive exams!
Example Trick: For 4 coins, number of outcomes = \( 2^4 = 16 \).
Tricks like these are taught in Vedantu’s live classes and used by toppers for fast calculation.
Try These Yourself
- Write the sample space for rolling two dice.
- List all outcomes for drawing a red card from a deck.
- If you toss a coin and pick a number from 1-3, what is the sample space?
- Find the number of outcomes when picking 2 balls from a bag of 5 colored balls.
Frequent Errors and Misunderstandings
- Confusing an event with the whole sample space (event is a subset; sample space is the entire set).
- Forgetting to count all combinations (like TT, HT, TH for two coins).
- Missing outcomes or duplicating them in sequences.
Relation to Other Concepts
The idea of sample space connects closely with topics such as Types of Events in Probability and Probability. Mastering sample space helps you solve all sorts of probability questions—whether in exams or while playing games involving chance.
Classroom Tip
A quick way to remember sample space: Always use curly brackets “{ }” and make your outcome lists or tables neat. For big experiments, draw a tree diagram or use smart formulas like \( 2^n \). Vedantu’s teachers often use Venn diagrams and color-coded tables to make learning visual and easy to remember in live classes.
We explored sample space in Maths—from definition, formula, examples, common mistakes, fast tricks, and how it connects with events and probability theory. Continue practicing sample space questions on Vedantu to become confident in solving all types of probability problems quickly and accurately.
- Probability and Statistics Symbols
- Sample Space Worksheet
- Probability Distribution
- Set Theory Symbols
FAQs on Sample Space in Maths: Full Guide with Examples
1. What is sample space in probability?
In probability, the sample space (often denoted as S) is the set of all possible outcomes of a random experiment. It's the complete collection of results that could occur.
2. What is the sample space for tossing a coin?
The sample space for tossing a single coin is S = {H, T}, where H represents heads and T represents tails.
3. What is the sample space for tossing two coins?
When tossing two coins, the sample space is S = {HH, HT, TH, TT}. Each outcome represents the result of the first and second coin toss, respectively.
4. How do you find the sample space for rolling a die?
The sample space for rolling a standard six-sided die is S = {1, 2, 3, 4, 5, 6}, representing the numbers that can appear on the top face.
5. What is the sample space for rolling two dice?
The sample space for rolling two dice involves ordered pairs. There are 36 possible outcomes: S = {(1,1), (1,2), ..., (6,6)}. Each pair (x,y) represents the outcome of the first and second die, respectively.
6. What is the difference between an event and a sample space?
The sample space is the set of *all* possible outcomes. An event is a specific subset of the sample space—a particular outcome or collection of outcomes you are interested in. For example, in rolling a die, the sample space is {1,2,3,4,5,6}, while an event might be 'rolling an even number' ({2,4,6}).
7. What is a discrete sample space?
A discrete sample space is one where the number of possible outcomes is finite or countably infinite. Examples include coin tosses, dice rolls, or the number of cars passing a point on a road in an hour.
8. What is a continuous sample space?
A continuous sample space contains an uncountably infinite number of outcomes, typically represented by an interval on the real number line. Examples include measuring height or temperature, or the time until an event occurs.
9. How do I use sample space to calculate probability?
Probability is calculated as the ratio of the number of favorable outcomes (belonging to the event) to the total number of outcomes in the sample space. P(Event) = (Number of favorable outcomes) / (Total number of outcomes in the sample space)
10. What are some ways to represent a sample space?
Sample spaces can be represented using various methods, including:
- Lists: e.g., {H, T}
- Sets: e.g., {1, 2, 3, 4, 5, 6}
- Tables: Useful for visualizing outcomes with multiple events.
- Tree diagrams: Especially helpful for visualizing sequential events.
11. Can the sample space be empty?
No, a sample space cannot be empty. By definition, it must contain at least one possible outcome, even if that outcome is an impossible event.
12. How does the size of the sample space affect probability calculations?
A larger sample space generally means a smaller probability for any given event (unless the number of favorable outcomes increases proportionally). The sample space forms the denominator in probability calculations, so its size directly impacts the result.
